Well, I’m not sure I agree that there is no need to learn this at all. As a matter of fact, I know I don’t agree. I think it is important to understand the meaning behind the algorithmic processes. I also think it is important not to focus only on these processes. Has your son been doing any problem solving that involves operations with decimals? Have there been any investigations into the patterns involved?

My concern with the message in your post is twofold: First, practice is necessary – but not if it is practice without understanding. Second: the number of people who responded in your comments about disliking/not understanding math disturbed me.

I actually liked your “sort of” method. That type of estimation skill is very important and I wish it were taught/discussed more. Asking if solutions are reasonable is a good problem solving technique whether one is using a calculator or not. This is one skill I think needs to be emphasized in any math curriculum.

I agree with Jackie. Learning algorithms not just for mastering the process but also understanding what is happening is very important — and both sides of the coin here are essential. Calculators don’t make math go away.

I also agree that the comments on that post were just depressing. We’ve got a serious problem with kids not knowing math in this country today, and the problem I am convinced is largely cultural and not curricular. I keep hoping that parents will stop saying “Oh gosh how I hate math” and instead say “My kid’s got to learn math so I had better suck it up myself!”

Long division in particular is a valuable algorithm because it provides extended practice with estimation, multiplication, and subtraction, all (pardon the word) “core” skills.

That being said, there can be a real problem with teaching the algorithm without making sense of it. I would advocate that the equivalent of: (and I bet my spacing turns out a mess, and I won’t bother to LaTeX it)

Anne: there is more to learning mathematics than just whether or not there will be calculators around – part of the process of learning is to get a gut feeling for the subject. To learn to instinctively double check any arithmetic you encounter in real life against the parts that went into it and to recognize when things simply are way wrong on sight.

As for the latter example, sure, if you get around to punching the numbers into your calculator, you’ll spot it. However, the point is that with a decent grounding, you will be able to reason about the amounts occurring instinctively and without using the calculator – so that even if you wouldn’t have cared enough to get the machinery out, you might have had alarms going off anyway.

To me, the most disturbing part of the My Tiny Kingdom post is the palpable fear of those who commented. This one by MamaD4 pretty well sums it up:

**My idea of hell would be a place that’s really hot where a creepy math teacher drones on and on for eternity, writing out long and complex equations on the chalkboard, and then assigns page after page of hideous problems like the one you just described.**

Now, there is something wrong when these people (who have all done this level of arithmetic when they were at school) are scared stiff of it. And there is even more of a problem when the calculus graduates cannot remember any calculus.

I’m with you, Jackie in your comment about estimation skills. That is something that should be retained (and promoted) in any curriculum review…

And there is even more of a problem when the calculus graduates cannot remember any calculus.

Hey, I resemble that statement. 😉

Seriously, I have a pure math degree from a well-respected school, and I went way beyond calculus, but I remember virtually none of it, because I don’t need it. What I remember, what I benefit from every day, are the thinking and problem-solving skills that I gained while studying all that stuff. And I have confidence that if I ever needed it, it would come back to me a lot more quickly.

But why is it a priori a problem when people forget about things like Calculus if they’re not using it? And why is that a “bigger” problem than people forgetting basic computation and numeracy skills?

In almost every (compulsory) math class I ever taught, the students would ask ‘Why are we doing this stuff? What is it good for? We will never use this stuff.’

Schools mostly get kids to ‘do math because it is good for them’, but by failing to show the students why it is good for them (I mean using applications that have meaning for the students), we produce a community of math phobes. And that community spreads the message to students that math is hard, math is useless and we all wish math would go away.

Apparently you chose to do higher math out of interest and still gained a lot, even though you don’t use the actual math content now. Where my comments are coming from is in consideration of the millions who must take math and who suffer without knowing what any of it is for.

But you’re right – my ranking is skewed. It is a bigger problem when people cannot do basic computations.

Mathmom – I think you are the exception. I also think that you probably remember more calc than you give yourself credit for – what does finding a derivative mean? I bet you know, even if you don’t remember the “steps” for doing so. I think Zac is referring to those who take math after high school and have no understanding of what they’re “learning”. This reminds me of many of the people in my math & math ed courses (the lack of understanding or desire to do so was frightening. I often heard, “Why do I need to know this, I’m only going to teach Algebra and Geometry”).

Zac – “suffer” aptly describes what most people feel when they think of math. I’m heading over to check out your poll, thanks for the link.

Jackie,
Derivative is rate of change; integral is area under the curve. I know that to find the integral you add up slices that in the limit have zero width. I have no clue how you find a derivative, but I know that “the integral of x is x squared over two.” One of my dad’s geek friends taught me to recite it on a camping trip when I was 5. I came back some 12 years later to tell him he forgot to tell me about the constant. 😉 And I know that if you take a velocity equation and differentiate it, you get an acceleration equation (rate of change of velocity!), though I didn’t know that when I memorized the stupid equations in physics the year before I took calculus! So, yeah, I still know the very basics. But if I needed to actually *use* calculus for anything, I’d need a very big brush up!

Zac,
I am a parent volunteer who helps teach math. I’m not trained to do so — trial by fire! But this is my 6th year, so I’m getting the hang of it. I teach problem solving. Hard problems. Contest problems. Today I made my middle schoolers, among other things, find the sum of the digits of 3333333334 squared. (Solve a Simpler Problem!) We all understand that that problem, or even one very like it, is never going to crop up in real life. But I hope that I’ve convinced them, and their parents, that what I’m teaching them does matter — that these problem solving skills, ways of thinking about hard problems, will serve them whenever they have hard problems to solve, even outside of mathematics.

So, Zac, what would (do?) you tell them — all those people who must take calculus and don’t appreciate why it is important?

Jackie: You’ll be happy to know that the “sort of like” method is now what they teach in the earlier years. My 3rd grade twins (more boys) learn to add from left to right instead of right to left as we did, and they check themselves by estimation to see if the answer is reasonable.

THAT makes a huge amount of sense to me.

The 6th grade curriculum, however, is a bit more mysterious. I’m a lawyer and a writer, so other than cooking I don’t do much math, but I can’t for the life of me figure out why they’re graphing inequalities.

Good on you for being a parent volunteer math teacher – I’m impressed!

A key aspect (if I understood you correctly) is that you are helping students who have volunteered for the math competitions that you are preparing them for. (Well, the reality is probably that their parents did the volunteering, but essentially it is not a compulsory activity.) In such a situation, the students are more likely to appreciate and enjoy having their horizons expanded.

Sadly, it’s different in school.

“So, Zac, what would (do?) you tell them — all those people who must take calculus and don’t appreciate why it is important?”

I make sure that calculus is not solely an exercise in moving letters and numbers around. From the beginning, I emphasise the historical motivations behind calculus and also get the students to explore simple applications like throwing balls, etc.

Zac, I do not just coach kids who chose math competitions. I teach every single kid in our little K-8 school. (That’s only about 30 kids.) I work with the “middle school” group for two sessions in a row, every other week. I work with the younger kids (a K-2 group and a 3-5 group) once every 2 weeks. (With the youngest kids in that younger group, I don’t really start on problem solving until there is some basic numeracy, but I actually do start problem solving with even the 5yos by about halfway through their first year.)

This is not a gifted school, it is a private progressive ungraded school. It is made up of kids whose parents care a lot about their education, enough to pay for private school most of us can’t really “afford”. So, it’s certainly not a typical group in that way, but the kids run the gamut of abilities and interests. Some of the kids are gifted, others struggle, and of course many are somewhere in between.

I teach mainly “problem solving” with a little mental math mixed in. I use the Figure it Out curriculum with the elementary kids, along with other material I find online, in other books, etc. I use the MOEMS materials with the upper elementary group and the middle school group and I use MATHCOUNTS materials with the middle school group, plus again other stuff I find online, in books, and on Dave Marain’s blog (yeah, I know that overlaps with online, but it deserves special mention because it’s so awesome). We give the MOEMS middle school contest during math class for the middle school students, so they are all required to take that one. We offer the MOEMS elementary level contest for the younger kids (which includes some of the kids in the middle school group, because “elementary” goes up to 6th grade), but no one is required to take that contest. But they are still all required to participate in the problem solving lessons I do. I also take interested students to the MATHCOUNTS competition. Again, they don’t have to do that, but they still have to participate in learning problem solving!

Many times the students complain at first that the math I teach is “too hard”. But eventually they all experience that aha! moment when they “get” a problem that looked “too hard” and it’s a great feeling, increases their confidence in their own skills, etc. When these kids go to high school, they know more than decimals and fractions, ratios and percents, and how to solve for x. They know how to think. They know how to struggle with a problem without giving up after 10 seconds. And they know the feeling of satisfaction that comes from working hard at something and succeeding.